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In the multiplication table for $\mathbb{Z^*_7}$, each row has all the elements of the set

My questions are:

  1. is each element a generator here? if not why?
  2. is an element considered as generator only if $Order(G) = Order(g)$ i.e $Order(g) = 6$ hence $g=3$ is a generator.
  3. is there any formula to find the number of generators?
  4. Some website says phi(n) gives the order of group some says $\varphi(n)$ gives the number of generators some says $\varphi(n)$ gives the number of elements with a multiplicative inverse. Which of these is correct?
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  1. is each element a generator here? if not why?

Take the $2$ and calculate $2 \cdot 2 \cdot 2 \equiv 8 \equiv 1 \mod7$. Therefore, not every element is a generator.

  1. is an element considered as generator only if $Order(G) = Order(g)$ i.e $Order(g) = 6$ hence $g=3$ is a generator.

A generator must generate all the element. If the order of $Order(a) = x < g = Order(G)$ then it cannot generates all the elements. $a^x =1 $ and the cyclic sub-group generated by $x$ contains only $x$ elements:

$$a^1,a^2, \ldots, a^x = 1$$

  1. is there any formula to find the number of generators?

No, there is no formula.

  1. Some website says $\varphi(n)$ gives the order of group some says $\varphi(n)$ gives the number of generators some says $\varphi(n)$ gives the number of elements with a multiplicative inverse. Which of these is correct?

Remember the $\mathbb{Z_7}$ is a field.

  • $\varphi(n)$ gives the order of the multiplicative group. In your case $\varphi(7) = 7-1 = 6$
  • 2 and 4 are not generators therefore $\varphi(n)$ cannot give the number of generators.
  • Being a field implies every element is invertible, thus $\varphi(7)$ gives the number of invertible elements, also for the composite case $-\mathbb{Z_n}-$, where $n$ is a composite number, $\varphi(n)$, gives the number of invertible elements.
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  • $\begingroup$ Take the 2 and calculate 2⋅2⋅2≡8≡1mod7. Therefore, not every element is a generator , could you explain this . why are we checking mod 7 of 8 ? $\endgroup$ – PDHide Dec 1 '18 at 17:36
  • $\begingroup$ if you had checked the table i posted , it can be observed that each row recreates the elements of z7 , then why isnt each element not considered as a generator $\endgroup$ – PDHide Dec 1 '18 at 17:38
  • $\begingroup$ Because we are talking about $\mathbb{Z_7}$. Each row contains distinct numbers since if there are more then one for a number say $a$ and $x$ is multiple times occurs then $x = a \cdot b= a \cdot c \Rightarrow b = c$. $\endgroup$ – kelalaka Dec 1 '18 at 17:42

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